Use partitioned matrices to prove by induction that for n = 2, 3,…,the n × n matrix A sh...

Use partitioned matrices to prove by induction that for n = 2, 3,…,the n × n matrix A shown below is invertible and B is its inverse.

For the induction step, assume A and B are (k + 1) × (k + 1) matrices, and partition A and B in a form similar to that displayed in Exercise 23.

Exercise 23:

Use partitioned matrices to prove by induction that the product of two lower triangular matrices is also lower triangular. [Hint: A(k + 1) × (k + 1) matrix A₁ can be written in the form below, where a is a scalar, v is in ℝ^k , and A is a k × k lower triangular matrix. See the Study Guide for help with induction.]